I will post a challenging math problem each day. The level of difficulty will vary, but most problems should not require any specialized knowledge beyond calculus. Problems are also posted on Twitter using the hashtag #mathpotd.
Sunday, September 20, 2009
Limit with nth roots
Evaluate the limit of (a^(1/n)+b^(1/n)-1)^n as n approaches infinity, where a, b > 1.
The answer is ab.
ReplyDeletea^(1/n) = e^(ln(a)/n) = 1 + ln(a)/n + O(1/n^2)
a^(1/n) + b^(1/n) - 1 = 1 + ln(a)/n + ln(b)/n + O(1/n^2)
a^(1/n) + b^(1/n) - 1 = 1 + ln(ab)/n + O(1/n^2)
(a^(1/n) + b^(1/n) - 1)^n = (1 + ln(ab)/n)^n + O(1/n)
= exp(ln(ab)) + O(1/n)
= ab + O(1/n)